Greenberger-Horne-Zeilinger state

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In physics, in the area of quantum information theory, a Greenberger-Horne-Zeilinger state is a certain type of entangled quantum state.

[edit] Definition

The GHZ state is an entangled quantum state in an M dimensions:

$|GHZ\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}.$

Most notably the 3-qubit GHZ state is: $|GHZ\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.$

[edit] Properties

Apparently there is no standard measure of multi-partite entanglement, but many measures define the GHZ to be maximally entangled.

Another important property of the GHZ state is that when we trace over one of the three systems

which is a mixed state.

On the other hand, if we were to measure any of subsystems we will leave behind either $|00\rangle$ or $|11\rangle$ which are not entangled. Thus, we say that the GHZ is maximally entangled. This is unlike the W state which leaves bipartite entanglements even when we measure one of its subsystems.

The GHZ state leads to striking non-classical correlations (1989). They can be easily shown to invalidate the ideas of Einstein (see EPR Paradox). This is an amplification of the Bell's theorem. The correlations can be utilized in some quantum information tasks. These include multipartner quantum cryptography (1998) and communication complexity tasks (1997, 2004).